How do you show that

Question

2sinx(cosx+cos3x)=sin4x?

Answer 1

2sinx(cos3x + cosx)

Recall : Trigonometry formulas cos3x = 4cos³x - 3cosx

Substitute cos3x = 4cos³x - 3cosx

2sinx(4cos³x - 3cosx + cosx)

2sinx(4cos³x - 2cosx)

2sinx(2cosx(2cos²x - 1))

Recall : trigonometry formulas sin2x = 2sinxcosx

2sin2x(2cos²x - 1)

Recall : cos2x = cos²x - sin²x = 2cos²x - 1

Substitute 2cos²x - 1 = cos2x

2sin2xcos2x

Sustitute 2sin2xcos2x = sin4x

sin4x.